Reading data¶
Textinho explicando o que foi feito no primeiro notebook
[ ]:
FILE_PATH = 'resampled_files/RESAMPLED_0101086161_20070516T060226.csv'
[ ]:
import pandas as pd
data = pd.read_csv(FILE_PATH)
data.head()
| DATE | WHITEFLUX | |
|---|---|---|
| 0 | 54236.757582 | 112521.329834 |
| 1 | 54236.767013 | 112758.045853 |
| 2 | 54236.776445 | 112943.042225 |
| 3 | 54236.785876 | 112562.266242 |
| 4 | 54236.795308 | 112789.303079 |
[ ]:
import numpy as np
time = data.DATE.to_numpy()
flux = data.WHITEFLUX.to_numpy()
[ ]:
from utils import *
[ ]:
curve = lightcurve.LightCurve(time, flux)
curve.plot()
Ideal Lowpass Filter - Traduzir¶
Um filtro bi-dimensional passa-baixa que deixa passar todas as frequências em um círculo de raio \(D_0\) a partir da origem e remove todas as frequências fora desse círculo é chamado de filtro passa-baixa ideal (ILPF) e é descrito como
onde \(D_0\) é uma constante positiva, e \(D(u)\) é a distância entre um ponto \(u\) até o centro do retângulo de frequência, ou seja, é definido por
sendo \(P\) o tamanho do vetor original preenchido (padded).
O ponto de transição entre \(H(u) = 1\) e \(H(u) = 0\) é chamado de frequência de corte
[ ]:
def ideal_filter(array, fourier_transform, cutoff_freq):
n_time = len(array)
D0 = cutoff_freq * n_time
for i in range(len(fourier_transform)):
if fourier_transform[i] > D0:
fourier_transform[i] = 0
y_filtered = np.real(np.fft.ifft(fourier_transform))
y_filtered += (array.mean() - y_filtered.mean())
return y_filtered
Plotting some results¶
[ ]:
filtered = curve.ideal_lowpass_filter(0.1)
filtered.view_filter_results()
[ ]:
filtered = curve.ideal_lowpass_filter(0.6)
filtered.view_filter_results()
Saving filtered data¶
[ ]:
cutoff_freqs = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9]
[ ]:
PATH_DIR = 'C:/Users/guisa/Desktop/filters/ideal/cutoffFreq0'
[ ]:
%time
for cutoff_freq in cutoff_freqs:
pass
# Saving filtered data
lightcurve.export_results_csv(WHERE_TO_SAVE_PATH=PATH_DIR+str(int(cutoff_freq*10)), filter_technique='ideal', cutoff_freq=cutoff_freq, order=None, numNei=None)
Wall time: 0 ns
Gaussian Lowpass Filter¶
The transfer function of a Gaussian 1-D lowpass filter (GLPFs) is defined by
where \(D(u)\) and \(D_0\) was defined on Eq. (1).
Note. The cutoff frequency must be given in Nyquist.
[ ]:
def gaussian_array(array, fourier_transform, cutoff_freq):
# Extrating information of the signal
n_time = len(array)
D0 = cutoff_freq * n_time
xc = n_time
# Creating the filter array
len_filter = len(fourier_transform)
filter = np.zeros(len_filter)
for i in range(len_filter):
filter[i] = exp( (-(i-(xc-1.0))**2)/(2*((D0 * n_time)**2)) )
return filter
Plotting some results - Completar¶
Note. It was observed that, at low cutoff frequencies (0.1 and 0.2 Nyquist), there is an effect in which the filtered curve undergoes a vertical displacement, as if a constant were added to the curve. This will not affect the modeling of the curves, but to prevent this effect from causing disturbances in the visualization of the results, the following operation is perfomed:
\[Flux\space Filtered\space += [mean(Raw\space Flux) - mean(Flux\space Filtered)]\]
[ ]:
filtered = curve.gaussian_lowpass_filter(cutoff_freq=0.1)
filtered.view_filter_results()
[ ]:
filtered = curve.gaussian_lowpass_filter(cutoff_freq=0.4)
filtered.view_filter_results()
Saving filtered data¶
[ ]:
cutoff_freqs = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9]
[ ]:
PATH_DIR = 'C:/Users/guisa/Desktop/filters/gaussian/cutoffFreq0'
[ ]:
%time
for cutoff_freq in cutoff_freqs:
# Saving filtered data
lightcurve.export_results_csv(WHERE_TO_SAVE_PATH=PATH_DIR+str(int(cutoff_freq*10)), filter_technique='gaussian', cutoff_freq=cutoff_freq, order=None, numNei=None)
Wall time: 0 ns
Butterworth Lowpass Filter¶
The transfer function of a Butterworth 1-D lowpass filter (BLPF) of order \(n\), and with cutoff frequency at a distance \(D_0\) from the origin, is defined as
where \(D(u)\) and \(D_0\) was defined on Eq. (1).
By the definition, the Butterworth filter have two free parameters: the cutoff frequency and the filtering order. Then, we can modify both, as we can see on the code cell below, intending to have the best results possibles.
Note. The cutoff frequency must be given in Nyquist.
[ ]:
def butterworth_array(array, fourier_transform, cutoff_freq, order):
# Extrating information of the signal
n_time = len(array)
D0 = cutoff_freq * n_time
xc = n_time
# Creating the filter array
len_filter = len(fourier_transform)
filter = np.zeros(len_filter)
for i in range(len_filter):
filter[i] = 1.0 / (1.0+(abs(i-(xc-1.0))/D0)**(2.0*order))
return filter
Plotting some results¶
[ ]:
filtered = curve.butterworth_lowpass_filter(2, 0.1)
filtered.view_filter_results()
[ ]:
filtered = curve.butterworth_lowpass_filter(4, 0.3)
filtered.view_filter_results()
Saving filtered data¶
[ ]:
orders = [1, 2, 3, 4, 5, 6]
cutoff_freqs = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9]
[ ]:
%time
for order in orders:
for cutoff_freq in cutoff_freqs:
# Saving filtered data
PATH_DIR = 'C:/Users/guisa/Desktop/filters/butterworth/order'+str(order)+'/cutoffFreq0'+str(int(cutoff_freq*10))
lightcurve.export_results_csv(WHERE_TO_SAVE_PATH=PATH_DIR, filter_technique='butterworth', cutoff_freq=cutoff_freq, order=order, numNei=None)
Wall time: 0 ns
Bessel Lowpass Filter¶
The transfer function, \(H(s)\), of a Bessel lowpass filter is defined by
where
and
Showing how it works…¶
Parameters
[ ]:
order = 2
cutoff_freq = 0.6
Control lib
[ ]:
from control import *
[ ]:
### Computing ak
from math import factorial
coef = []
i = 0
while i <= order:
ak = (factorial(2*order - i)) / ( 2**(order - i)*factorial(i)*factorial(order - i) )
# print(ak)
coef.append(ak)
i += 1
print(coef)
[3.0, 3.0, 1.0]
[ ]:
### Computing θn(s)
s = TransferFunction.s
theta_array = []
k = 0
for k in range(order+1):
theta_n = coef[k] * (s**k)
theta_array.append(theta_n)
# numerical_numerator = coef[0]
# print(theta_n)
print(theta_array[0])
print(theta_array[1])
print(theta_array[2])
3
-
1
3 s
---
1
s^2
---
1
[ ]:
### Computing H(s)
coef_numerator = theta_array[0]
list_denominator = theta_array[:]
[ ]:
denominator = 0
for item in list_denominator:
denominator += item
print(denominator)
s^2 + 3 s + 3
-------------
1
[ ]:
### Filling in transfer function
G = coef_numerator / denominator
print(G)
print(type(G))
3
-------------
s^2 + 3 s + 3
<class 'control.xferfcn.TransferFunction'>
Applying
[ ]:
def bessel(array, fourier_transform, cutoff_freq, order):
# Extracting features from signal
n_time = len(array)
D0 = cutoff_freq * n_time
xc = n_time
# Creating the bessel transfer function array
len_filter = len(fourier_transform)
filter = np.zeros(len_filter)
i=0
for i in range(len_filter):
filter[i] = np.real(evalfr(G, ( np.abs(i-(xc-1.0))/D0 )))
return filter
Plotting some results¶
[ ]:
filtered = curve.bessel_lowpass_filter(2, 0.1, numExpansion=100)
filtered.view_filter_results()
[ ]:
filtered = curve.bessel_lowpass_filter(4, 0.3, numExpansion=100)
filtered.view_filter_results()
Saving filtered data¶
[ ]:
orders = [1, 2, 3, 4, 5, 6]
cutoff_freqs = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9]
[ ]:
%time
for order in orders:
for cutoff_freq in cutoff_freqs:
# Saving filtered data
PATH_DIR = 'C:/Users/guisa/Desktop/filters/bessel/order'+str(order)+'/cutoffFreq0'+str(int(cutoff_freq*10))
lightcurve.export_results_csv(WHERE_TO_SAVE_PATH=PATH_DIR, filter_technique='bessel', cutoff_freq=cutoff_freq, order=order, numNei=None)
Median Filter¶
O filtro mediano, por sua vez, é aplicado de uma forma consideravelmente diferente, na qual cada valor dos dados filtrados corresponde a uma mediana de um grupo de valores adjacentes nos dados originais. Esse filtro já é utilizado com certa frequência em análises de curvas de luz.
[ ]:
from scipy.signal import medfilt
def median_filter(array, window_size):
return medfilt(array, window_size)
Reference: Scipy Documentation
Plotting some results¶
[ ]:
filtered = curve.median_filter(3)
filtered.view_filter_results()
[ ]:
filtered = curve.median_filter(9)
filtered.view_filter_results()
Saving filtered data¶
[ ]:
neighbors = [3, 5, 7, 9, 11]
[ ]:
%time
for neighbor in neighbors:
# Saving filtered data
PATH_DIR = 'C:/Users/guisa/Desktop/filters/median/numNei'+str(int(neighbor))
lightcurve.export_results_csv(PATH_DIR, 'median', cutoff_freq=None, order=None, numNei=neighbor)
Wall time: 0 ns
API¶
Here, you can apply all the filtering processes, with whatever parameters you want, on whatever lightcurve
Select the Lightcurve:¶
RESAMPLED_0100725706_20070516T060226
RESAMPLED_0101086161_20070516T060226
RESAMPLED_0101206560_20070516T060226
RESAMPLED_0101368192_20070516T060050
RESAMPLED_0102671819_20071023T223035
RESAMPLED_0102671819_20120112T183055
RESAMPLED_0102708694_20071023T223035
RESAMPLED_0102708694_20120112T183055
RESAMPLED_0102725122_20071023T223035
RESAMPLED_0102725122_20120112T183055
RESAMPLED_0102764809_20071023T223035
RESAMPLED_0102890318_20070206T133547
RESAMPLED_0102912369_20070203T130553
RESAMPLED_0105118236_20100708T204534
RESAMPLED_0105209106_20080415T231048
RESAMPLED_0105228856_20100408T223049
RESAMPLED_0105793995_20080415T231048
RESAMPLED_0105819653_20080415T231048
RESAMPLED_0105833549_20080415T231048
RESAMPLED_0105891283_20080415T231048
RESAMPLED_0106017681_20080415T231048
RESAMPLED_0110839339_20081116T190224
RESAMPLED_0110864907_20081116T190224
RESAMPLED_0221686194_20081011T143035
RESAMPLED_0300001097_20081116T190224
RESAMPLED_0310247220_20090403T220030
RESAMPLED_0311519570_20090403T220030
RESAMPLED_0315198039_20100305T001525
RESAMPLED_0315211361_20100305T001525
RESAMPLED_0315239728_20100305T001525
RESAMPLED_0630831435_20110708T151253
RESAMPLED_0652180928_20110708T151253
RESAMPLED_0652180991_20110708T151253
[4]:
LIGHTCURVE = 'RESAMPLED_0110839339_20081116T190224'
Filtering processes availables:¶
Ideal (Cutoff frequency)
Gaussian (Cutoff frequency)
Butterworth (Order, Cutoff frequency)
Bessel (Order, Cutoff frequency, numExpansion=100)
Median (Number of neighbors)
[2]:
from utils import *
import pandas as pd
import numpy as np
[5]:
data = pd.read_csv('https://raw.githubusercontent.com/Guilherme-SSB/IC-CoRoT_Kepler/main/resampled_files/' + LIGHTCURVE + '.csv')
time = data.DATE.to_numpy()
flux = data.WHITEFLUX.to_numpy()
curve = lightcurve.LightCurve(time, flux)
curve.plot()
[ ]:
help(curve.how_to_filter)
Help on method how_to_filter in module utils.lightcurve:
how_to_filter(order: int, cutoff_freq: float, numNei: int, numExpansion: int) method of utils.lightcurve.LightCurve instance
This function describes how to filtering using this library
Parameters
----------
order : int
Used in Butterworth and Bessel filtering. Matches the filter order.
cutoff_freq : float
Used in Ideal, Gaussian, Butterworth and Bessel filtering. Matches the cutoff frequency.
numNei : int
Used in Median. Matches the number of neighbors to consider
numExpansion : int
Used in all processes. Corresponds to how much you want to expanded the curve's edges
(to avoid some problems caused by the Fast Fourier Transform algorithm).
Preliminary tests show that all processes works fine with numExpansion=70,
except for Bessel filtering which required numExpansion=100
Methods
-------
curve.ideal_lowpass_filter(cutoff_freq)
curve.gaussian_lowpass_filter(cutoff_freq)
curve.butterworth_lowpass_filter(order, cutoff_freq)
curve.bessel_lowpass_filter(order, cutoff_freq)
curve.median_filter(numNei)
Examples
--------
>>> from utils import *
>>> curve = lightcurve.Lightcurve(time=[1, 2, 3, 4], flux=[10, 15, 12, 20])
>>> print(curve)
<utils.lightcurve.LightCurve object>
>>> curve.plot()
>>> filtered = curve.bessel_lowpass_filter(order=3, cutoff_freq=0.4, numExpansion=100)
>>> print(filtered)
<utils.lightcurve.FilteredLightCurve object>
>>> filtered.view_filter_results()
[8]:
filtered = curve.butterworth_lowpass_filter(order=2, cutoff_freq=0.3)
filtered.view_filter_results()